Shortest Path in Binary Matrix

You’re given a binary matrix, and you need to find the shortest path from the top-left corner to the bottom-right corner, using only cells with a value of 0, and moving in all 8 possible directions. If no such path exists, return -1.

A good approach to solve this problem is to use Breadth-First Search (BFS). Here’s the code:

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from collections import deque

class Solution:
    def shortestPathBinaryMatrix(self, grid: List[List[int]]) -> int:
        n = len(grid)
        if grid[0][0] == 1 or grid[n - 1][n - 1] == 1:
            return -1

        directions = [(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)]
        queue = deque([(0, 0, 1)])  # x, y coordinates and length of path so far

        while queue:
            x, y, path_len = queue.popleft()

            # Check if reached destination
            if x == y == n - 1:
                return path_len

            # Explore all 8 directions
            for dx, dy in directions:
                nx, ny = x + dx, y + dy
                if 0 <= nx < n and 0 <= ny < n and grid[nx][ny] == 0:
                    queue.append((nx, ny, path_len + 1))
                    grid[nx][ny] = 1  # Mark as visited

        return -1

This code first checks if the start or end points are blocked and returns -1 if so. Then it initializes a queue and iteratively explores the 8 neighboring cells from each cell, marking them as visited by setting them to 1. If it reaches the destination, it returns the path length; if no valid path is found, it returns -1.

Identifying Problem Isomorphism

“Shortest Path in Binary Matrix” involves finding the shortest path from the top-left corner to the bottom-right corner of a grid (treated as a graph), where you can move in eight directions (not just left/right/up/down). The problem involves using a breadth-first search (BFS) algorithm to find the shortest path.

This problem is similar in nature to “01 Matrix” (LeetCode #542). In “01 Matrix”, given a matrix consists of 0 and 1, find the distance of the nearest 0 for each cell. This problem also uses BFS to find the shortest distance to a particular type of node (0 in this case).

The difference between these two problems is the goal of the search: in “Shortest Path in Binary Matrix” we’re looking for the shortest path to a specific location (the bottom-right corner), while in “01 Matrix” we’re looking for the shortest path to a specific type of node (a 0). However, both problems require similar approaches, use the BFS algorithm, and involve traversing a matrix.

Thus, these two problems are not isomorphic in the sense that they are exactly the same, but they share enough characteristics to be considered approximately isomorphic, in that they use the same type of algorithm to solve a similar type of problem.

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class Solution:
    def shortestPathBinaryMatrix(self, grid: List[List[int]]) -> int:
        # check if source and target are not clear cells
        if grid[0][0] != 0 or grid[-1][-1] != 0:
            return -1

        N = len(grid)            
        # offsets required for all 8 directions
        offsets = [(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)]

        q = deque()
        q.append((0,0)) # starting point
        visited = {(0, 0)}

        # finds unvisited clear cells using 8 offsets
        def get_neighbours(x,y):
            for x_offset, y_offset in offsets:
                new_row = x + x_offset
                new_col = y + y_offset

                if 0 <= new_row < N and 0 <= new_col < N and not grid[new_row][new_col] and (new_row, new_col) not in visited:
                    yield (new_row, new_col)

        current_distance = 1 # start with one clear cell
        # standard iterative BFS traversal
        while q:
            length = len(q)

            # loop through all the cells at the same distance
            for _ in range(length):
                row, col = q.popleft()

                if row == N-1 and col==N-1: # reached target
                    return current_distance

                # loop though all valid neignbours
                for p in get_neighbours(row, col):
                    visited.add(p)
                    q.append(p)
     
            current_distance+=1 # update the level or distance from source

        return -1

Problem Classification

Given an n x n binary matrix grid, return the length of the shortest clear path in the matrix. If there is no clear path, return -1.

A clear path in a binary matrix is a path from the top-left cell (i.e., (0, 0)) to the bottom-right cell (i.e., (n - 1, n - 1)) such that:

All the visited cells of the path are 0. All the adjacent cells of the path are 8-directionally connected (i.e., they are different and they share an edge or a corner). The length of a clear path is the number of visited cells of this path.

Example 1:

Input: grid = [[0,1],[1,0]] Output: 2

Example 2:

Input: grid = [[0,0,0],[1,1,0],[1,1,0]] Output: 4 Example 3:

Input: grid = [[1,0,0],[1,1,0],[1,1,0]] Output: -1

Constraints:

n == grid.length n == grid[i].length 1 <= n <= 100 grid[i][j] is 0 or 1

Language Agnostic Coding Drills

  1. Understanding the Problem Statement: Recognizing the problem as a shortest path search in a binary matrix. The shortest path could be identified using a breadth-first search (BFS) approach.

  2. Arrays and Lists: Understanding how to declare and manipulate lists and arrays.

  3. Conditionals: Understanding how to use if conditions to make decisions in code. Here, it is used to check if the start and end of the grid are clear.

  4. 2D Arrays/Grids: Understanding how to work with 2D arrays or grids, which are a list of lists in Python.

  5. Working with Tuples: Learning to create, access and manipulate tuples which are used to represent points on the grid.

  6. Creating Helper Functions: Understanding how to create helper functions, in this case, get_neighbours() to find all valid neighboring cells.

  7. Understanding Breadth-First Search (BFS): Learning the algorithm and how it can be used to find the shortest path in a grid or graph.

  8. Working with Queues: Understanding how to use a queue data structure, which is needed for the BFS algorithm.

  9. While Loops: Learning to use while loops to keep running the BFS until all possibilities have been exhausted.

  10. For Loops: Understanding for loops to iterate through the queue and each cell’s neighbors.

  11. Understanding Yield Statement: Understanding the yield statement in Python, which returns a generator from a function that can be iterated over.

  12. Updating Data Structures: Learning how to update sets and queues as the BFS progresses.

  13. Understanding Offsets: Understanding how to use offsets to explore different directions in the grid.

  14. Return Statements: Learning to use return statements to provide the output of the function.

  15. Complexity Analysis: Understanding the time complexity of the algorithm (which is O(N^2) for a grid of size NxN).

These concepts build on each other. Starting with simple array and list manipulations, the learner gradually moves to complex topics like BFS, generators and time complexity analysis. Combining these elements will lead to the final solution.

Targeted Drills in Python

  1. Understanding the Problem Statement

    • Problem: Write a function that takes in a grid (a 2D list of integers) and returns a statement if the start and end of the grid are clear (0).
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    def is_clear(grid):
    if grid[0][0] == 0 and grid[-1][-1] == 0:
        return "Start and End are clear"
    else:
        return "Start and/or End are not clear"

  2. Arrays and Lists

    • Problem: Create a list of integers and perform basic operations like accessing elements, updating elements, and finding the length of the list.
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    my_list = [1, 2, 3, 4, 5]
print(my_list[0])  # access the first element
my_list[1] = 7  # update the second element
print(len(my_list))  # find the length of the list

  3. Conditionals

    • Problem: Write a function that takes an integer and returns whether it is positive, negative, or zero.
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    def check_number(n):
    if n > 0:
        return "Positive"
    elif n < 0:
        return "Negative"
    else:
        return "Zero"

  4. 2D Arrays/Grids

    • Problem: Create a 2D list (grid), access elements and sublists, and find the dimensions of the grid.
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    grid = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
print(grid[0])  # access the first sublist
print(grid[1][2])  # access the third element of the second sublist
rows = len(grid)
columns = len(grid[0]) if grid else 0  # get the number of columns from the first row
print("Rows:", rows, "Columns:", columns)

  5. Working with Tuples

    • Problem: Create a tuple, access its elements, and try to modify an element (which should raise an error because tuples are immutable).
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    my_tuple = (1, 2, 3)
print(my_tuple[0])  # access the first element
# my_tuple[1] = 7  # this line would raise a TypeError

  6. Creating Helper Functions

    • Problem: Create a helper function that squares an integer, then use it in another function that squares all integers in a list.
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    def square(n):
    return n ** 2

def square_all(numbers):
    return [square(n) for n in numbers]

  7. Understanding Breadth-First Search (BFS)

    • Problem: Given a binary tree, perform a level-order traversal (BFS) and print the value of each node.
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    from collections import deque

class Node:
    def __init__(self, val, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

def bfs(root):
    queue = deque([root])
    while queue:
        node = queue.popleft()
        print(node.val)
        if node.left:
            queue.append(node.left)
        if node.right:
            queue.append(node.right)

  8. Working with Queues

    • Problem: Create a queue (using Python’s deque), enqueue some integers, and then dequeue them in the same order.
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    from collections import deque

queue = deque()
queue.append(1)  # enqueue 1
queue.append(2)  # enqueue 2
print(queue.popleft())  # dequeue and print 1
print(queue.popleft())  # dequeue and print 2

  9. While Loops

    • Problem: Write a function that prints the integers from 1 to 10 using a while loop.
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    def print_to_ten():
    i = 1
    while i <= 10:
        print(i)
        i += 1

  10. For Loops

    • Problem: Write a function that prints the square of each integer from 1 to 10 using a for loop.
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    def print_squares():
    for i in range(1, 11):
        print(i ** 2)

  11. Understanding Yield Statement

    • Problem: Write a generator function that yields each integer from 1 to a given number.
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    def up_to(n):
    i = 1
    while i <= n:
        yield i
        i += 1

for number in up_to(10):
    print(number)

  12. Updating Data Structures

    • Problem: Create a set of integers, add a new integer, and remove an existing integer.
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    my_set = {1, 2, 3}
my_set.add(4)  # add 4 to the set
my_set.remove(2)  # remove 2 from the set
print(my_set)

  13. Understanding Offsets

    • Problem: Given a point (a tuple of x and y coordinates), use an offset to move the point to a new position.
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    point = (1, 2)
offset = (3, -1)
new_point = (point[0] + offset[0], point[1] + offset[1])
print(new_point)

  14. Return Statements

    • Problem: Write a function that takes two integers and returns their sum.
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    def add(a, b):
    return a + b

  15. Complexity Analysis

    • Problem: Given a function that iterates over a list of n integers, describe its time complexity in big O notation (O(n)).
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    def print_all(numbers):
    for number in numbers:
        print(number)
# Time complexity is O(n) because we are performing a single operation for each element in the list.

By practicing these drills, you will become familiar with the basic and complex concepts required to solve the original problem.

10 Prerequisite LeetCode Problems

The problem involves finding the shortest path in a binary matrix, which can be solved using breadth-first search (BFS) or depth-first search (DFS) algorithms, along with other concepts related to traversing a matrix. Here are some LeetCode problems of lesser complexity that can help you prepare:

  1. Easy Difficulty:

    • 733. Flood Fill: This problem requires you to implement a simple DFS on a matrix, which is a fundamental concept for the given problem.
    • 542. 01 Matrix: This problem can help you practice BFS on a matrix.

  2. Medium Difficulty:

    • 200. Number of Islands: This problem also requires DFS or BFS traversal in a matrix, similar to the given problem.
    • 130. Surrounded Regions: This problem requires understanding how to traverse a matrix using DFS.
    • 994. Rotting Oranges: This problem involves a grid and a process that spreads over time, which can be solved using a BFS approach.
    • 127. Word Ladder: Although it’s not a matrix problem, it’s about finding the shortest transformation sequence, which can be solved using BFS, similar to the given problem.

  3. Hard Difficulty:

These problems should provide a good foundation for understanding the concepts needed to tackle the ‘Shortest Path in Binary Matrix’ problem.

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Problem Classification

Given an n x n binary matrix grid, return the length of the shortest clear path in the matrix. If there is no clear path, return -1. A clear path in a binary matrix is a path from the top-left cell (i.e., (0, 0)) to the bottom-right cell (i.e., (n - 1, n - 1)) such that: All the visited cells of the path are 0. All the adjacent cells of the path are 8-directionally connected (i.e., they are different and they share an edge or a corner). The length of a clear path is the number of visited cells of this path.

Example 1:

Input: grid = [[0,1],[1,0]] Output: 2

Example 2:

Input: grid = [[0,0,0],[1,1,0],[1,1,0]] Output: 4

Example 3:

Input: grid = [[1,0,0],[1,1,0],[1,1,0]] Output: -1

Constraints:

n == grid.length n == grid[i].length 1 <= n <= 100 grid[i][j] is 0 or 1

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the edge cases?

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method.

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

  1. Parameters:

    • What are the inputs to the method?
    • What types are these parameters?
    • What do these parameters represent in the context of the problem?

  2. Preconditions:

    • Before this method is called, what must be true about the state of the program or the values of the parameters?
    • Are there any constraints on the input parameters?
    • Is there a specific state that the program or some part of it must be in?

  3. Method Functionality:

    • What is this method expected to do?
    • How does it interact with the inputs and the current state of the program?

  4. Postconditions:

    • After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
    • What does the return value represent or indicate?
    • What side effects, if any, does the method have?

  5. Error Handling:

    • How does the method respond if the preconditions are not met?
    • Does it throw an exception, return a special value, or do something else?

Problem Decomposition

  1. Problem Understanding:

    • Can you explain the problem in your own words? What are the key components and requirements?

  2. Initial Breakdown:

    • Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?

  3. Subproblem Refinement:

    • For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?

  4. Task Identification:

    • Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?

  5. Task Abstraction:

    • For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?

  6. Method Naming:

    • Can you give each task a simple, descriptive name that makes its purpose clear?

  7. Subproblem Interactions:

    • How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

  1. Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.

  2. Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?

  3. If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.

  4. If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?

  5. Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.

  6. Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

  1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

  2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

  3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

Similar Problems

Can you suggest 10 problems from LeetCode that require similar problem-solving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem.